April 21, 2023 Jonathan Valvano EE345M/EE380L.6

11. Use FFT to Design DF

Lab 4 Application of RTOS

• Input sound, analog filter • Digital filter, FFT• Display amplitude versus freq on the oLED

Noise Reduction

-2048

-1536

-1024

-512

0

512

1024

1536

2048

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Time

Ou

tpu

t

Reference EE345L book, chapter 15

Lab4 Spectrum Analyzer

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

Design a linear FIR digital filter

• Specify gain versus frequency

• Specify phase versus frequencyFIR digital filter

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Frequency

Gai

n

y(n)x(n)

Fast Fourier Transform (FFT)

• FFT is a faster version of the Discrete Fourier Transform (DFT)

• FFT spectrum of a cosine, which has a frequency of 0.1 Fs– Fs = 10000 Hz– Cosine freq = 1000 Hz

• Region interested in from 0 to Fs/2 (symmetric about Fs/2)

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

FIR digital filter

• Y(z) = H(z) X(z)

• h(n) = IFFT {H(z)}

• Convolution

– y(n) = h(n)*x(n)

• Constants h0, h1, ... hN-1

• y(n) = h0·x(n)+h1·x(n-1)+ ... +hN-1·x(n-(N-1))

• N multiplies, N-1 additions per sample

FIR digital filter

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Frequency

Gai

n

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

How to choose sampling rate• Nyquist Rate

• Limitation of display

• Limitation of processor

• Limitation of RAM

• Limitation of human eyes and ears

• Limitation of communication channel

aliasedproperly represented

undetectable

frequency

Z2

z

f s1 2

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

How to choose number of samples

• Frequency resolution = fs/N

• Increase in N results in better frequency resolution

• However, increase in N leads to a bigger MACQ buffer and more multiplies and additions

• Does not need to be a power of 2–DFT calculated once, off line

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

Design process (1)

• Specify desired gain and phase, 0 to ½ fs

– k goes from 0 to N/2 (f = k fs/N)

– H(k) is complex– |H(k)| is gain– angle(H(k)) is phase

• For ½ fs to fs

– H(N-k) is complex conjugate of H(k) – Poles and zeros are in complex conjugate pairs

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

Design process (2)

• Take IDFT of H(k) to yield h(n)– n goes from 0 to N-1– h(n) will be real, because H(k) symmetric

• The digital filter is– y(n) = h0·x(n)+h1·x(n-1)+ ... +hN-1·x(n-(N-1))

Binary Fixed point notation

• Q formats• Qn Number (16 bit)

– n: specifies no of bits used for precision

– 16-n: no of bits used for Range

• Eg: 96.450 (unsigned number) - What Qn?

• How is this number stored as an integer?

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

Example

• Open FIRdesign51.xls

• Change sampling rate to 10,000 Hz

• Adjust red desired gain to make BPF– Pass 2 to 4 kHz– Look at sharp corner versus round corner

• Notice linear phase

• Copy 51 coefficients into software

2k to 4k BPF51-term FIR filter

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Frequency (Hz)

Gai

n (V

/V)

Actual

Desired

FIR Filter SW designconst long h[51]={-3,-9,4,5,0,17,5,-20,-5,-7,-22, 24,41,-8,2,1,-74,-31,71,20,33,125,-119,-350,67, 462,67,-350,-119,125,33,20,71,-31,-74,1,2,-8,41, 24,-22,-7,-5,-20,5,17,0,5,4,-9,-3};static unsigned int n=50; // 51,52,… 101short Filter(short data){unsigned int k;static short x[102]; // this MACQ needs twiceshort y;n++; if(n==102) n=51; x[n] = x[n-51] = data; // two copies of new data y = 0; for(k=0;k<51;k++){ y = y + h[k]*x[n-k]; // convolution } y = y/256; // fixed point return y;}

Circular Buffering

Source: “Communication system design using DSP algorithms” by Steven A. Tretter (chapter 3, page 73)

April 21, 2023 Jonathan Valvano EE345M/EE380L.6

Optimization

• Pointer implementations of MACQ faster

• Do not try and shift the data

• Convolution x[n]*h[n] takes N multiplies, N-1 additions per sample– Can be optimized to N/2 multiplies– Coefficients are symmetric

• Assembly optimization with MLA– Multiply with accumulate